Aims

Here, I will analyse the HVPG dataset, evaluate aspects of its test-retest reliability, and perform a power analysis for comparing treatments.

Libraries

library(tidyverse)
library(relfeas)
library(readxl)
library(hrbrthemes)
library(extrafont)
library(lme4)
library(lmerTest)
library(effsize)
library(broom)
library(RColorBrewer)
library(knitr)
library(cowplot)
library(progress)
library(kableExtra)
library(perm)
library(ggbeeswarm)
library(psych)
library(permuco)

extrafont::loadfonts(quiet = T)

theme_set(theme_ipsum_rc())
nsims <- 1e4
overwrite <- FALSE

knitr::opts_chunk$set(fig.path = "figures/", dev="png", dpi=600,
                      warning=FALSE, message=FALSE)

set.seed(42)

Data

trt_tidy <- read_excel("../RawData/TEST_RETEST_FINA_DATABASE_TIDY.xlsx") %>% 
  mutate(author = str_match(Description, "(^\\w*)")[,2]) %>% 
  select(-contains("≦"))

trt_wide <- read_excel("../RawData/TEST_RETEST_FINA_DATABASE.xlsx") %>% 
  select(New_Description, `Serial number`) %>% 
  select(-contains("≦"))

trt_studydemog <- read_excel("../RawData/FINAL_AGGREGATED_ICCs-2.xlsx") %>% 
  select(Study, Perc_Alc, Perc_Decomp, Days=TIMEDAYS, 
         Centre = CENTER,
         n_Patients = NUMBEROFPATIENTs) %>% 
  mutate(Centre = ifelse(Centre == 1, yes = "Multi-centre", "Single-centre"))

Now let’s add the new new description to the trt_tidy sheet

trt_tidy <- trt_tidy %>% 
  left_join(trt_wide)

Study Names

studynames <- trt_tidy %>% 
  select(Study = New_Description, 
         Technique = `Technique - Balloon/ catheter`) %>% 
  unique() %>% 
  mutate(Technique = ifelse(Technique=="Wedged Catheter",
                            yes = "Wedged Catheter",
                            no = "Balloon-tipped Catheter")) %>% 
  mutate(Technique = ifelse(is.na(Technique),
                            yes = "Balloon-tipped Catheter",
                            no = Technique)) %>% 
  mutate(Catheter = ifelse(Technique=="Wedged Catheter",
                           yes="Wedged", no="Balloon tip"))

Test-Retest Analysis

As if one study

Now, we look at the data as if it were all one study, however we also divide by whether the study contains decompensated patients.

trt_all <- trt_tidy %>% 
  filter(Description != "Spahr. Octreotide") %>%
  select(-Study) %>%
  rename(Study= New_Description) %>% 
  left_join(trt_studydemog) %>% 
  mutate(decomp = ifelse(Perc_Decomp >0 , 
                           yes="Includes Decompensated",
                           no = "Only Compensated")) %>%
  group_by(decomp) %>% 
  nest() %>% 
  mutate(trt = map(data, ~relfeas::trt(data = .x, 
                                       values='PP', 
                                       cases = "Serial number", 
                                       rater = 'MEASUREMENT' )$tidy)) %>% 
  select(-data) %>% 
  unnest(trt)
  

trt_all
kable(trt_all, digits=2)
decomp mean sd cv skew kurtosis icc icc_l icc_u wscv sdd absvar signvar signvar_sd
Includes Decompensated 17.79 4.86 0.27 0.05 -0.49 0.82 0.77 0.86 0.12 5.68 0.12 0 0.16
Only Compensated 16.20 3.95 0.24 0.38 -0.54 0.87 0.84 0.90 0.09 3.92 0.10 0 0.12

So, overall, we estimate that our smallest detectable difference in an individual is 3.9mmHg for only compensated patients, and 5.7mmHg for studies including decompensated patients

trt_all_detailed <- trt_tidy %>% 
  filter(Description != "Spahr. Octreotide") %>%
  select(-Study) %>%
  rename(Study= New_Description) %>% 
  left_join(trt_studydemog) %>% 
  mutate(decomp = ifelse(Perc_Decomp >0 , 
                           yes="Includes Decompensated",
                           no = "Only Compensated")) %>%
  filter(Description != "Spahr. Octreotide") %>% 
  group_by(decomp) %>% 
  nest() %>% 
  mutate(trt = map(data, ~relfeas::trt(data = .x, 
                                       values='PP', 
                                       cases = "Serial number", 
                                       rater = 'MEASUREMENT' )))

trt_all_detailed$decomp[1]
## [1] "Includes Decompensated"
trt_all_detailed$trt[[1]]$sdd
## $value
## [1] 5.679483
## 
## $lbound
## [1] 5.067141
## 
## $ubound
## [1] 6.461497
trt_all$sdd_l <- trt_all_detailed$trt[[1]]$sdd$lbound
trt_all$sdd_u <- trt_all_detailed$trt[[1]]$sdd$ubound

trt_all_detailed$decomp[2]
## [1] "Only Compensated"
trt_all_detailed$trt[[2]]$sdd
## $value
## [1] 3.924009
## 
## $lbound
## [1] 3.525756
## 
## $ubound
## [1] 4.424489
trt_all$sdd_l[2] <- trt_all_detailed$trt[[2]]$sdd$lbound
trt_all$sdd_u[2] <- trt_all_detailed$trt[[2]]$sdd$ubound

trt_all_detailed$decomp[1]
## [1] "Includes Decompensated"
trt_all_detailed$trt[[1]]$sddm
## $value
## [1] 0.3192501
## 
## $lbound
## [1] 0.2808643
## 
## $ubound
## [1] 0.3636064
trt_all$sddm   <- trt_all_detailed$trt[[1]]$sddm$value*100
trt_all$sddm_l <- trt_all_detailed$trt[[1]]$sddm$lbound*100
trt_all$sddm_u <- trt_all_detailed$trt[[1]]$sddm$ubound*100

trt_all_detailed$decomp[2]
## [1] "Only Compensated"
trt_all_detailed$trt[[2]]$sddm
## $value
## [1] 0.2422477
## 
## $lbound
## [1] 0.2151506
## 
## $ubound
## [1] 0.2731478
trt_all$sddm[2]   <- trt_all_detailed$trt[[2]]$sddm$value*100
trt_all$sddm_l[2] <- trt_all_detailed$trt[[2]]$sddm$lbound*100
trt_all$sddm_u[2] <- trt_all_detailed$trt[[2]]$sddm$ubound*100

Now, let’s prepare this as a table for below the forest plot

overall_n <- map_dbl(trt_all_detailed$data, nrow)

overall <- trt_all %>% 
  ungroup() %>% 
  rename(Study = decomp) %>% 
  mutate(decomp = "Overall",
         Catheter = "Balloon tip",
         n = overall_n) %>% 
  arrange(desc(Study)) %>% 
  mutate(Study = ifelse(Study=="Includes Decompensated",
                        "Includes Decompensated*",
                        Study))

overall

Grouped by study

trt_study <- trt_tidy %>% 
  group_by(New_Description) %>% 
  nest() %>% 
  mutate(outcomes = map(data, ~ trt( data=.x, values='PP', 
                                     cases = "Serial number", rater = 'MEASUREMENT' )))

trt_study <- trt_study %>% 
  mutate(n = map_dbl(data, nrow))
  

tidytrt <- map_df(trt_study$outcomes, 'tidy') %>% 
  mutate(n = trt_study$n) %>% 
  mutate(Study = trt_study$New_Description) %>% 
  select(Study, everything()) %>% 
  left_join(studynames) %>% 
  left_join(trt_studydemog) %>% 
  mutate(decomp = ifelse(Perc_Decomp >0 , 
                           yes="Includes Decompensated",
                           no = "Only Compensated")) %>% 
  mutate(decomp = fct_inorder(decomp)) %>% 
  select(-n_Patients)


knitr::kable(tidytrt, digits = 2)
Study mean sd cv skew kurtosis icc icc_l icc_u wscv sdd absvar signvar signvar_sd n Technique Catheter Perc_Alc Perc_Decomp Days Centre decomp
Abraldes 2008 (D) 20.28 4.43 0.22 0.12 -1.00 0.81 0.62 0.91 0.10 5.43 0.11 -0.03 0.14 36 Balloon-tipped Catheter Balloon tip 44.0 100.0 30.00 Multi-centre Includes Decompensated
Abraldes 2008 (C) 18.47 2.91 0.16 0.28 -1.47 0.81 0.45 0.94 0.07 3.65 0.08 0.02 0.10 16 Balloon-tipped Catheter Balloon tip 44.0 0.0 30.00 Multi-centre Only Compensated
Albillos 1995 19.60 3.82 0.20 0.14 -1.69 0.94 0.82 0.98 0.05 2.74 0.05 0.01 0.07 20 Balloon-tipped Catheter Balloon tip 60.0 0.0 90.00 Single-centre Only Compensated
Berzigotti 2010 18.62 2.21 0.12 0.05 -2.35 0.96 0.42 1.00 0.03 1.55 0.04 0.01 0.06 4 Balloon-tipped Catheter Balloon tip 50.0 0.0 16.00 Single-centre Only Compensated
Blei 1987 15.06 4.67 0.31 0.14 -1.53 0.97 0.85 0.99 0.06 2.49 0.06 0.05 0.07 18 Balloon-tipped Catheter Balloon tip 100.0 100.0 0.04 Single-centre Includes Decompensated
Debernardi 2007 14.59 1.76 0.12 0.49 -0.56 0.64 0.32 0.83 0.07 3.01 0.08 0.05 0.10 34 Balloon-tipped Catheter Balloon tip 21.7 0.0 365.00 Single-centre Only Compensated
Jayakumar 2013 22.72 3.11 0.14 0.18 -0.90 0.62 0.10 0.88 0.09 5.42 0.11 0.00 0.13 16 Balloon-tipped Catheter Balloon tip 75.0 100.0 56.00 Multi-centre Includes Decompensated
Kimer 2017 16.18 4.46 0.28 0.69 0.10 0.63 0.32 0.82 0.17 7.62 0.19 -0.02 0.25 36 Balloon-tipped Catheter Balloon tip 72.2 100.0 28.00 Single-centre Includes Decompensated
Lebrec 2012 18.04 3.01 0.17 -1.10 -0.06 0.66 0.06 0.92 0.10 5.01 0.13 -0.06 0.14 12 Balloon-tipped Catheter Balloon tip 50.0 0.0 0.04 Multi-centre Only Compensated
Merkel 2004 12.39 1.18 0.10 0.12 -0.26 0.87 0.63 0.96 0.04 1.22 0.04 0.02 0.05 18 Balloon-tipped Catheter Balloon tip 57.7 0.0 730.00 Multi-centre Only Compensated
Moller 2000 15.38 5.45 0.35 -0.41 -1.51 0.96 0.87 0.99 0.07 3.10 0.07 0.02 0.11 16 Balloon-tipped Catheter Balloon tip 100.0 87.5 0.02 Single-centre Includes Decompensated
Reverter 2015 15.76 4.48 0.28 0.39 -1.06 0.95 0.90 0.98 0.06 2.76 0.07 -0.03 0.08 42 Balloon-tipped Catheter Balloon tip 47.6 0.0 15.00 Multi-centre Only Compensated
Schepke 2001 18.25 3.85 0.21 0.06 -1.18 0.69 0.42 0.85 0.12 6.01 0.12 0.00 0.17 36 Balloon-tipped Catheter Balloon tip 72.5 100.0 7.00 Single-centre Includes Decompensated
Spahr 2007 17.69 2.96 0.17 -0.07 -1.32 0.14 -0.47 0.67 0.16 7.64 0.21 -0.07 0.22 16 Wedged Catheter Wedged 62.5 75.0 90.00 Single-centre Includes Decompensated
Schwarzer 2017 20.50 4.86 0.24 -0.06 -1.50 0.94 0.83 0.98 0.06 3.39 0.06 0.02 0.09 20 Balloon-tipped Catheter Balloon tip 80.0 80.0 28.00 Single-centre Includes Decompensated
Pomier 1987 15.42 5.50 0.36 1.45 0.52 0.95 0.54 0.99 0.09 3.71 0.09 0.10 0.08 12 Balloon-tipped Catheter Balloon tip 62.5 100.0 172.50 Single-centre Includes Decompensated
Hidaka 2011 14.75 3.71 0.25 0.27 -0.87 0.79 0.59 0.90 0.12 4.80 0.13 -0.06 0.16 38 Balloon-tipped Catheter Balloon tip 16.7 0.0 0.01 Multi-centre Only Compensated
McCormick 1992 17.44 4.40 0.25 -0.18 -0.32 0.94 0.87 0.97 0.06 3.04 0.08 0.02 0.09 40 Balloon-tipped Catheter Balloon tip 75.0 75.0 0.01 Single-centre Includes Decompensated
Pozzi 2005 16.11 3.61 0.22 0.10 -1.14 0.78 0.41 0.93 0.11 4.89 0.14 0.08 0.14 18 Balloon-tipped Catheter Balloon tip 0.0 100.0 182.50 Single-centre Includes Decompensated
Garcia-Tsao 2020 (C) 16.63 4.00 0.24 0.24 -0.50 0.82 0.72 0.88 0.10 4.75 0.13 0.01 0.15 100 Balloon-tipped Catheter Balloon tip 0.0 0.0 168.00 Multi-centre Only Compensated
Garcia-Tsao 2020 (D) 16.32 4.73 0.29 -0.76 -0.15 0.26 -0.19 0.72 0.27 12.17 0.35 -0.25 0.31 14 Balloon-tipped Catheter Balloon tip 0.0 100.0 168.00 Multi-centre Includes Decompensated
Fukada 2014 17.28 4.15 0.24 0.32 -0.83 0.93 0.77 0.98 0.07 3.19 0.07 0.02 0.10 16 Balloon-tipped Catheter Balloon tip 25.0 0.0 0.04 Single-centre Only Compensated

Now let’s make a table for the paper with the relevant things.

tidytrt_table <- tidytrt %>% 
  mutate(signvar_sd = signvar_sd * mean) %>% 
  select(decomp, 
         Study,
         n,
         "Decompensated (%)" = Perc_Decomp,
         "Alcoholic (%)" = Perc_Alc,
         "Mean Days Elapsed" = Days,
         Mean = mean, CV = cv, 
         WSCV = wscv, ICC = icc,
         SDD = sdd,
         "Change SD" = signvar_sd) %>% 
  arrange(desc(decomp), Study)

decomp_change <- head(
  which(
    tidytrt_table$decomp == tail(tidytrt_table$decomp, 1)), 1 )

tidytrt_kable <- knitr::kable(tidytrt_table[,-1], digits=2) %>% 
  kable_styling("striped", full_width = F) %>%
  pack_rows(head(tidytrt_table$decomp, 1), 1, decomp_change-1) %>%
  pack_rows(tail(tidytrt_table$decomp, 1), decomp_change, nrow(tidytrt_table))

tidytrt_kable
Study n Decompensated (%) Alcoholic (%) Mean Days Elapsed Mean CV WSCV ICC SDD Change SD
Only Compensated
Abraldes 2008 (C) 16 0.0 44.0 30.00 18.47 0.16 0.07 0.81 3.65 1.94
Albillos 1995 20 0.0 60.0 90.00 19.60 0.20 0.05 0.94 2.74 1.46
Berzigotti 2010 4 0.0 50.0 16.00 18.62 0.12 0.03 0.96 1.55 1.06
Debernardi 2007 34 0.0 21.7 365.00 14.59 0.12 0.07 0.64 3.01 1.40
Fukada 2014 16 0.0 25.0 0.04 17.28 0.24 0.07 0.93 3.19 1.71
Garcia-Tsao 2020 (C) 100 0.0 0.0 168.00 16.63 0.24 0.10 0.82 4.75 2.44
Hidaka 2011 38 0.0 16.7 0.01 14.75 0.25 0.12 0.79 4.80 2.37
Lebrec 2012 12 0.0 50.0 0.04 18.04 0.17 0.10 0.66 5.01 2.54
Merkel 2004 18 0.0 57.7 730.00 12.39 0.10 0.04 0.87 1.22 0.62
Reverter 2015 42 0.0 47.6 15.00 15.76 0.28 0.06 0.95 2.76 1.34
Includes Decompensated
Abraldes 2008 (D) 36 100.0 44.0 30.00 20.28 0.22 0.10 0.81 5.43 2.77
Blei 1987 18 100.0 100.0 0.04 15.06 0.31 0.06 0.97 2.49 1.06
Garcia-Tsao 2020 (D) 14 100.0 0.0 168.00 16.32 0.29 0.27 0.26 12.17 5.06
Jayakumar 2013 16 100.0 75.0 56.00 22.72 0.14 0.09 0.62 5.42 2.96
Kimer 2017 36 100.0 72.2 28.00 16.18 0.28 0.17 0.63 7.62 3.98
McCormick 1992 40 75.0 75.0 0.01 17.44 0.25 0.06 0.94 3.04 1.56
Moller 2000 16 87.5 100.0 0.02 15.38 0.35 0.07 0.96 3.10 1.67
Pomier 1987 12 100.0 62.5 172.50 15.42 0.36 0.09 0.95 3.71 1.26
Pozzi 2005 18 100.0 0.0 182.50 16.11 0.22 0.11 0.78 4.89 2.24
Schepke 2001 36 100.0 72.5 7.00 18.25 0.21 0.12 0.69 6.01 3.15
Schwarzer 2017 20 80.0 80.0 28.00 20.50 0.24 0.06 0.94 3.39 1.78
Spahr 2007 16 75.0 62.5 90.00 17.69 0.17 0.16 0.14 7.64 3.95 
save_kable(tidytrt_kable, file = "figures/tidytrt_kable.jpg")

ICC

Let’s have a look at the ICC as a measure of between-subject differentiability (i.e. reliability).

icc_out <- select(tidytrt, Study, icc, icc_l, icc_u, decomp, n) %>% 
  left_join(studynames) %>% 
  mutate(decomp = factor(decomp, levels=c(
    "Includes Decompensated", "Only Compensated"))) %>% 
  arrange(decomp, icc) %>% 
  bind_rows(overall) %>% 
  mutate(Study = fct_inorder(Study))

ICCs <- ggplot(icc_out,aes(x=icc,y=Study, 
                           colour=Catheter)) +
  #geom_rect(aes(xmin=0.8, xmax=1, ymin=-Inf, ymax=Inf),
  #          alpha = .1, fill="grey", colour="grey") +
  facet_grid(decomp~., scales="free", space="free") +
  geom_point(aes(size=log(n)), shape=18) + 
  scale_x_continuous(breaks = seq(0, 1, by=0.2))+
  geom_errorbarh(aes(xmax = icc_u, xmin = icc_l), height = 0.2) +
  #geom_vline(xintercept = 1, linetype = "longdash") +
  labs(y="", x="Intraclass Correlation Coefficient (ICC)") +
  theme(text = element_text(size=20)) +
  ggtitle("ICC (95% CI)") +
  scale_colour_manual(values = c("black", "red")) +
  coord_cartesian(xlim = c(-0.1, 1)) +
                  #ylim = c(0,12)) +
  guides(size = FALSE) +
  NULL

ICCs + theme(legend.position = "none")

SDD

Absolute

Let’s have a look at the study-by-study SDD, but in raw units

sdd_out <- map_df(trt_study$outcomes, "sdd") %>% 
  mutate(Study = trt_study$New_Description) %>% 
  rename(sdd=value, sdd_l = lbound, sdd_u = ubound) %>% 
  left_join(studynames) %>% 
  left_join(select(tidytrt, Study, decomp, n)) %>% 
  left_join(trt_studydemog) %>% 
  mutate(decomp = ifelse(Perc_Decomp >0 , 
                           yes="Includes Decompensated",
                           no = "Only Compensated")) %>% 
  mutate(decomp = factor(decomp, levels=c(
    "Includes Decompensated", "Only Compensated"))) %>% 
  arrange(decomp, desc(sdd)) %>% 
  bind_rows(overall) %>% 
  mutate(Study = fct_inorder(Study))

SDDs <- ggplot(sdd_out,aes(x=sdd,y=Study, 
                           colour=Catheter)) +
  facet_grid(decomp~., scales="free", space="free") +
  geom_point(aes(size=log(n)), shape=18) + 
  scale_x_continuous(breaks = seq(0, 20, by=2))+
  geom_errorbarh(aes(xmax = sdd_u, xmin = sdd_l), height = 0.15) +
  #geom_vline(xintercept = 1, linetype = "longdash") +
  labs(y="", x="Smallest Detectable Difference (mmHg)") +
  theme(text = element_text(size=20)) +
  ggtitle("SDD (95% CI)") +
  scale_colour_manual(values = c("black", "red")) +
  guides(colour=FALSE, shape=FALSE) +
  #scale_shape_manual(values = c(18, 19)) +
  coord_cartesian(xlim=c(0,16)) +
  guides(size = FALSE) +
  NULL

SDDs  #+ annotate("rect", xmin = 0.75, xmax = 1, ymin="Spahr 2007", ymax="Blei 1987", alpha = .2, fill="grey")

Percentage

Now the study-by-study SDD, but in percentages

sddm_out <- map_df(trt_study$outcomes, "sddm") %>% 
  mutate(Study = trt_study$New_Description) %>% 
  rename(sddm=value, sddm_l = lbound, sddm_u = ubound) %>% 
  mutate(sddm=100*sddm, sddm_l = 100*sddm_l, sddm_u = 100*sddm_u) %>% 
  left_join(studynames) %>% 
  left_join(select(tidytrt, Study, decomp, n)) %>% 
  left_join(trt_studydemog) %>% 
  mutate(decomp = ifelse(Perc_Decomp >0 , 
                           yes="Includes Decompensated",
                           no = "Only Compensated")) %>% 
  mutate(decomp = factor(decomp, levels=c(
    "Includes Decompensated", "Only Compensated"))) %>% 
  arrange(decomp, desc(sddm)) %>% 
  bind_rows(overall) %>% 
  mutate(Study = fct_inorder(Study))

SDDms <- ggplot(sddm_out,aes(x=sddm,y=Study, 
                           colour=Catheter)) +
  facet_grid(decomp~., scales="free", space="free") +
  geom_point(aes(size=log(n)), shape=18) + 
  scale_x_continuous(breaks = seq(0, 80, by=10))+
  geom_errorbarh(aes(xmax = sddm_u, xmin = sddm_l), height = 0.15) +
  #geom_vline(xintercept = 1, linetype = "longdash") +
  labs(y="", x="Smallest Detectable Difference (%mmHg)") +
  theme(text = element_text(size=20)) +
  ggtitle("%SDD (95% CI)") +
  scale_colour_manual(values = c("black", "red")) +
  guides(colour=FALSE, shape=FALSE) +
  #scale_shape_manual(values = c(18, 19)) +
  coord_cartesian(xlim=c(0,80)) +
  guides(size = FALSE) +
  NULL

SDDms  #+ annotate("rect", xmin = 0.75, xmax = 1, ymin="Spahr 2007", ymax="Blei 1987", alpha = .2, fill="grey")

Figures together

SDD_together <- cowplot::plot_grid(SDDs, SDDms, ncol = 2,
                                   labels = c("B", "C"),
                                   label_size = 30)

SDD_together

Alternatively, all in a row

ICCs_row <- ICCs + guides(colour=FALSE, shape=FALSE)

forest_legend <- get_legend(ICCs + 
                              theme(legend.position="bottom") +
                              scale_shape_discrete("Patients:") +
                              scale_colour_manual("Catheter:",
                                values=c("black", "red")))

forest_row_legendless <- plot_grid(ICCs_row, SDDs, SDDms,
          nrow=1, labels=c("A", "B", "C"), 
          label_size=30)

forest_row <- plot_grid(forest_row_legendless, forest_legend,
                        nrow=2, rel_heights = c(15,1))

forest_row

ggsave(forest_row, filename = "figures/forest_three_row.png", width = 16, height = 8)

Just ICC and SDD

forest_row2_legendless <- plot_grid(ICCs_row, SDDs,
          nrow=1, labels=c("A", "B"), 
          label_size=30)

forest_row2 <- plot_grid(forest_row2_legendless, forest_legend,
                        nrow=2, rel_heights = c(15,1))

forest_row2

Are they different

To test the difference between the groups, we’ll bootstrap the difference. That means we take 1000 random samples from each group of the same size with replacement, and calculate the difference to get the null distribution. Then we compare the difference we see to the bootstrap distribution.

trt_compare <- trt_tidy %>% 
  filter(Description != "Spahr. Octreotide") %>%
  select(-Study) %>%
  rename(Study= New_Description) %>% 
  left_join(trt_studydemog) %>% 
  mutate(decomp = ifelse(Perc_Decomp >0 , 
                           yes="Includes Decompensated",
                           no = "Only Compensated")) %>%
  group_by(decomp) %>% 
  nest()

ICC

bootstrap_icc_single <- function(data) {
  
  sample_ids <- unique(data$`Serial number`)
  ids <- sample(sample_ids, length(sample_ids), replace=TRUE)
  
  data_nested <- data %>% 
    select(`Serial number`, MEASUREMENT, PP) %>% 
    nest_by(`Serial number`)
  
  boot_sample <- tibble(
    `Serial number` = ids
  ) %>% 
    left_join(data_nested, by="Serial number") %>% 
    select(-`Serial number`) %>% 
    mutate(boot_serial = 1:n()) %>% 
    unnest(data) %>% 
    relfeas::trt_widify("PP", "boot_serial", "MEASUREMENT")

  boot_icc <- suppressMessages(
    suppressWarnings(
      psych::ICC(boot_sample[,c(2,3)])$results$ICC[2] ) )
  return(boot_icc)
}

bootstrap_icc <- function(n, data, colname=NULL) {
  
  out <- tibble(
    n = 1:n
  ) %>% 
    mutate(boot_icc = map_dbl(n, ~bootstrap_icc_single(data)))
  
  if(!is.null(colname)) {
    colnames(out)[2] <- colname
  }
  
  return(out)
}

That seems to work. Now let’s test it!

trt_compare_iccvals <- trt_compare %>% 
  mutate(boot = map2(data, decomp, ~bootstrap_icc(1000, .x))) %>% 
  select(-data) %>% 
  unnest(boot)
trt_compare_icc <- trt_compare_iccvals %>% 
  spread(decomp, boot_icc) %>% 
  mutate(dif = `Only Compensated` - `Includes Decompensated`)

quantile(trt_compare_icc$dif, c(0.025, 0.975)) # 95% CI
##        2.5%       97.5% 
## -0.02794468  0.14634069
1-sum(trt_compare_icc$dif > 0)/1000 # one-sided p value
## [1] 0.115

Not significant

SDD

bootstrap_sdd_single <- function(data) {
  
  sample_ids <- unique(data$`Serial number`)
  ids <- sample(sample_ids, length(sample_ids), replace=TRUE)
  
  data_nested <- data %>% 
    select(`Serial number`, MEASUREMENT, PP) %>% 
    nest_by(`Serial number`)
  
  boot_sample <- tibble(
    `Serial number` = ids
  ) %>% 
    left_join(data_nested, by="Serial number") %>% 
    select(-`Serial number`) %>% 
    mutate(boot_serial = 1:n()) %>% 
    unnest(data) %>% 
    relfeas::trt_widify("PP", "boot_serial", "MEASUREMENT")

  boot_sdd <- suppressMessages(
    suppressWarnings(
      agRee::agree.sdd(as.matrix(boot_sample[,c(2,3)]))$value ) )
  return(boot_sdd)
}

bootstrap_sdd <- function(n, data, colname=NULL) {
  
  out <- tibble(
    n = 1:n
  ) %>% 
    mutate(boot_sdd = map_dbl(n, ~bootstrap_sdd_single(data)))
  
  if(!is.null(colname)) {
    colnames(out)[2] <- colname
  }
  
  return(out)
}

bootstrap_sdd_single(trt_compare$data[[1]])
## [1] 4.638595
bootstrap_sdd(100, trt_compare$data[[1]], "test")

And now we test

trt_compare_sddvals <- trt_compare %>% 
  mutate(boot = map2(data, decomp, ~bootstrap_sdd(1000, .x))) %>% 
  select(-data) %>% 
  unnest(boot)
trt_compare_sdd <- trt_compare_sddvals %>% 
  spread(decomp, boot_sdd) %>% 
  mutate(dif = `Includes Decompensated` - `Only Compensated`)

quantile(trt_compare_sdd$dif, c(0.025, 0.975)) # 95% CI
##     2.5%    97.5% 
## 0.574844 3.019354
1-sum(trt_compare_sdd$dif > 0)/1000 # one-sided p value
## [1] 0.003

SDDM

Percentage SDD

bootstrap_sddm_single <- function(data) {
  
  sample_ids <- unique(data$`Serial number`)
  ids <- sample(sample_ids, length(sample_ids), replace=TRUE)
  
  data_nested <- data %>% 
    select(`Serial number`, MEASUREMENT, PP) %>% 
    nest_by(`Serial number`)
  
  boot_sample <- tibble(
    `Serial number` = ids
  ) %>% 
    left_join(data_nested, by="Serial number") %>% 
    select(-`Serial number`) %>% 
    mutate(boot_serial = 1:n()) %>% 
    unnest(data) %>% 
    relfeas::trt_widify("PP", "boot_serial", "MEASUREMENT")

  boot_sddm <- suppressMessages(
    suppressWarnings(
      agRee::agree.sddm(as.matrix(boot_sample[,c(2,3)]))$value ) )
  return(boot_sddm)
}

bootstrap_sddm <- function(n, data, colname=NULL) {
  
  out <- tibble(
    n = 1:n
  ) %>% 
    mutate(boot_sddm = map_dbl(n, ~bootstrap_sddm_single(data)))
  
  if(!is.null(colname)) {
    colnames(out)[2] <- colname
  }
  
  return(out)
}

bootstrap_sddm_single(trt_compare$data[[1]])
## [1] 0.2864259
bootstrap_sddm(100, trt_compare$data[[1]], "test")

And now we test

trt_compare_sddmvals <- trt_compare %>% 
  mutate(boot = map2(data, decomp, ~bootstrap_sddm(1000, .x))) %>% 
  select(-data) %>% 
  unnest(boot)
trt_compare_sddm <- trt_compare_sddmvals %>% 
  spread(decomp, boot_sddm) %>% 
  mutate(dif = `Includes Decompensated` - `Only Compensated`)

quantile(trt_compare_sddm$dif, c(0.025, 0.975)) # 95% CI
##        2.5%       97.5% 
## 0.005952583 0.148503260
1-sum(trt_compare_sddm$dif > 0)/1000 # one-sided p value
## [1] 0.014

Causes of differences

Here, we perform an exploratory analysis of which factors may contribute to differences

trt_wide <- trt_tidy %>% 
  select(-Study) %>%
  rename(Study= New_Description) %>% 
  spread(MEASUREMENT, PP) %>% 
  rename(Meas1 = `1`,
         Meas2 = `2`) %>% 
  mutate(change = Meas2 - Meas1,
         abschange=abs(change),
         meanval = (Meas1 + Meas2)/2) %>% 
  left_join(studynames) %>% 
  left_join(trt_studydemog) %>% 
  mutate(Description = as.factor(Description)) %>% 
  filter(Description != "Spahr. Octreotide") %>% 
  mutate(decomp = ifelse(Perc_Decomp == 0,
                         "Only Compensated",
                         "Includes Decompensated"))


trt_wide_c <- trt_wide %>% 
  filter(Perc_Decomp == 0)

trt_wide_dc <- trt_wide %>% 
  filter(Perc_Decomp > 0)

Skewness

Absolute

Let’s just visualise our distribution.

ggplot(trt_wide, aes(x=abschange)) +
  geom_density(fill="black",alpha=0.5)

And between groups

ggplot(trt_wide, aes(x=abschange)) +
  geom_density(aes(colour=decomp, fill=decomp),alpha=0.5) +
  facet_wrap(decomp~., nrow=2) +
  theme(legend.position="bottom")

And let’s get some values for that too

psych::describe(trt_wide$abschange)
psych::describeBy(trt_wide$abschange, group=trt_wide$decomp)
## 
##  Descriptive statistics by group 
## group: Includes Decompensated
##    vars   n mean   sd median trimmed  mad min  max range skew kurtosis   se
## X1    1 131 2.02 2.08    1.5    1.69 1.48   0 13.5  13.5 2.19     7.12 0.18
## ------------------------------------------------------------ 
## group: Only Compensated
##    vars   n mean   sd median trimmed  mad min max range skew kurtosis   se
## X1    1 150  1.5 1.33      1    1.35 1.48   0   6     6 0.87        0 0.11

Signed

Let’s just visualise our overall distribution.

ggplot(trt_wide, aes(x=change)) +
  geom_density(fill="black",alpha=0.5)

… and by group

ggplot(trt_wide, aes(x=change)) +
  geom_density(aes(colour=decomp, fill=decomp),alpha=0.5) +
  facet_wrap(decomp~., nrow=2) +
  theme(legend.position="bottom")

For signed differences, let’s assess whether they differ from zero.

summary(lmer(change ~ 1 + (1|Study), data=trt_wide))
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: change ~ 1 + (1 | Study)
##    Data: trt_wide
## 
## REML criterion at convergence: 1304.2
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -5.1470 -0.4712 -0.0170  0.5292  3.7968 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Study    (Intercept) 0.2793   0.5285  
##  Residual             5.8542   2.4195  
## Number of obs: 281, groups:  Study, 21
## 
## Fixed effects:
##             Estimate Std. Error       df t value Pr(>|t|)
## (Intercept) -0.03655    0.19379  9.51195  -0.189    0.854
summary(lmer(change ~ 1 + (1|Study), data=trt_wide_c))
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: change ~ 1 + (1 | Study)
##    Data: trt_wide_c
## 
## REML criterion at convergence: 635.4
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -2.44715 -0.52563 -0.01548  0.49067  2.95984 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Study    (Intercept) 0.07633  0.2763  
##  Residual             3.97149  1.9929  
## Number of obs: 150, groups:  Study, 10
## 
## Fixed effects:
##             Estimate Std. Error       df t value Pr(>|t|)
## (Intercept) -0.01245    0.19502  6.35033  -0.064    0.951
summary(lmer(change ~ 1 + (1|Study), data=trt_wide_dc))
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: change ~ 1 + (1 | Study)
##    Data: trt_wide_dc
## 
## REML criterion at convergence: 650.1
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.2173 -0.4228  0.0196  0.5096  3.2825 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Study    (Intercept) 0.7253   0.8517  
##  Residual             7.9327   2.8165  
## Number of obs: 131, groups:  Study, 11
## 
## Fixed effects:
##             Estimate Std. Error       df t value Pr(>|t|)
## (Intercept) -0.05436    0.36373  5.60384  -0.149    0.886

Not significantly different from zero in either the combined sample or each group.

And some values

psych::describe(trt_wide$change)
psych::describeBy(trt_wide$change, group=trt_wide$decomp)
## 
##  Descriptive statistics by group 
## group: Includes Decompensated
##    vars   n  mean   sd median trimmed  mad   min max range  skew kurtosis   se
## X1    1 131 -0.06 2.91      0    0.08 2.22 -13.5   9  22.5 -0.76     3.54 0.25
## ------------------------------------------------------------ 
## group: Only Compensated
##    vars   n mean   sd median trimmed  mad min max range skew kurtosis   se
## X1    1 150    0 2.01      0   -0.01 1.48  -5   6    11 0.13     0.11 0.16

Figures

Let’s make some combined figures for the paper

sign_change_distr <- trt_wide %>% 
  mutate(decomp = as.factor(decomp)) %>% 
  ggplot(aes(x=change, colour=decomp, fill=decomp, group=decomp)) +
  geom_density(aes(colour=decomp, fill=decomp),alpha=0.5) +
  theme(legend.position=c(0.3, 0.9)) +
  scale_color_brewer("Patients", type = "qual", palette = 1) +
  scale_fill_brewer("Patients", type = "qual", palette = 1) +
  guides(fill=FALSE) +
  labs(y = "",
       x = "Signed Change from first to second measurement (mmHg)") +
  theme(axis.text.y = element_blank(),
        axis.ticks.y=element_blank())

sign_change_distr

abs_change_distr <- trt_wide %>% 
  mutate(decomp = as.factor(decomp)) %>% 
  ggplot(aes(x=abschange, colour=decomp, fill=decomp, group=decomp)) +
  geom_density(aes(colour=decomp, fill=decomp),alpha=0.5) +
  theme(legend.position=c(0.5, 0.9)) +
  scale_color_brewer("Patients", type = "qual", palette = 1) +
  scale_fill_brewer("Patients", type = "qual", palette = 1) +
  guides(fill=FALSE) +
  labs(y = "",
       x = "Absolute Change from first to second measurement (mmHg)") +
  theme(axis.text.y = element_blank(),
        axis.ticks.y=element_blank())

abs_change_distr

Percentage Decomp

permTREND(formula=abschange ~ Perc_Decomp, data=trt_wide,
          method="exact.mc")
## 
##  Exact Permutation Test Estimated by Monte Carlo
## 
## data:  x and y
## p-value = 0.002
## alternative hypothesis: true correlation of x and y is not equal to 0
## sample estimates:
## correlation of x and y 
##              0.1772453 
## 
## p-value estimated from 999 Monte Carlo replications
## 99 percent confidence interval on p-value:
##  0.00000000 0.01057916
decomp_plot <- ggplot(trt_wide, aes(x=Perc_Decomp, y=abschange)) +
  geom_beeswarm(aes(colour=as.factor(Study), 
                    group=as.factor(Perc_Decomp)),
                alpha=0.4, cex=1) +
  guides(colour=FALSE) + 
  geom_smooth(method="lm", se = FALSE) +
  labs(x="Decompensated Patients (%)",
       y="Absolute Change (mmHg)")

decomp_plot

Days elapsed

permuco::lmperm(abschange ~ Days + Perc_Decomp, data=trt_wide)
## Table of marginal t-test of the betas
## Permutation test using freedman_lane to handle nuisance variables and 5000 permutations.
##               Estimate Std. Error t value parametric Pr(>|t|)
## (Intercept)  1.5273173   0.175545  8.7004           2.959e-16
## Days        -0.0004743   0.000719 -0.6597           5.100e-01
## Perc_Decomp  0.0060002   0.002299  2.6104           9.535e-03
##             permutation Pr(<t) permutation Pr(>t) permutation Pr(>|t|)
## (Intercept)                                                           
## Days                    0.2686             0.7316               0.5006
## Perc_Decomp             0.9974             0.0028               0.0080

Now let’s plot, after correcting for the effect of the patient groups.

correct_for_decomp <- function(formula) {
  
  formula <- as.formula(formula)
  
  coefficients <- coef(permuco::lmperm(formula, 
                                       data=trt_wide))

  predicted <- as.character(formula[2])
  
  after_decomp_corr <- trt_wide[[predicted]] - 
    trt_wide[["Perc_Decomp"]] * coefficients[which(names(coefficients)=="Perc_Decomp")]
  
  return(after_decomp_corr)
  
}

days_plot <- trt_wide %>% 
  mutate(abschange_dccorr = correct_for_decomp(abschange ~ Days + Perc_Decomp)) %>% 
  ggplot(aes(x=Days, y=abschange_dccorr)) +
  geom_beeswarm(aes(colour=Study, group=Study), alpha=0.4, cex=1) +
  guides(colour=FALSE) + 
  geom_smooth(method="lm", se=FALSE) +
  labs(y="Absolute Change (mmHg)\n(after correction for Decompensated Percentage)",
       x="Days Elapsed between Measurements")

days_plot

Percentage Alcoholic

permuco::lmperm(abschange ~ Perc_Alc + Perc_Decomp, data=trt_wide)
## Table of marginal t-test of the betas
## Permutation test using freedman_lane to handle nuisance variables and 5000 permutations.
##             Estimate Std. Error t value parametric Pr(>|t|) permutation Pr(<t)
## (Intercept)  1.94300   0.167265  11.616           1.080e-25                   
## Perc_Alc    -0.01849   0.003803  -4.862           1.944e-06              2e-04
## Perc_Decomp  0.01364   0.002546   5.358           1.767e-07              1e+00
##             permutation Pr(>t) permutation Pr(>|t|)
## (Intercept)                                        
## Perc_Alc                 1e+00                2e-04
## Perc_Decomp              2e-04                2e-04
permTREND(formula=abschange ~ Perc_Alc, data=trt_wide_c,
          method="exact.mc")
## 
##  Exact Permutation Test Estimated by Monte Carlo
## 
## data:  x and y
## p-value = 0.002
## alternative hypothesis: true correlation of x and y is not equal to 0
## sample estimates:
## correlation of x and y 
##             -0.2785993 
## 
## p-value estimated from 999 Monte Carlo replications
## 99 percent confidence interval on p-value:
##  0.00000000 0.01057916
permTREND(formula=abschange ~ Perc_Alc, data=trt_wide_dc,
          method="exact.mc")
## 
##  Exact Permutation Test Estimated by Monte Carlo
## 
## data:  x and y
## p-value = 0.002
## alternative hypothesis: true correlation of x and y is not equal to 0
## sample estimates:
## correlation of x and y 
##              -0.276011 
## 
## p-value estimated from 999 Monte Carlo replications
## 99 percent confidence interval on p-value:
##  0.00000000 0.01057916
perc_alc_plot <- trt_wide %>% 
  mutate(abschange_dccorr = correct_for_decomp(abschange ~ Perc_Alc + Perc_Decomp)) %>% 
  ggplot(aes(x=Perc_Alc, y=abschange_dccorr)) +
  geom_beeswarm(aes(colour=Study, group=Study), alpha=0.4, cex=1) +
  guides(colour=FALSE) + 
  geom_smooth(method="lm", se=FALSE) +
  labs(y="Absolute Change (mmHg)\n(after correction for Decompensated Percentage)",
       x="Percentage of Alcoholic Patients (%)")

perc_alc_plot

Multicentre

permuco::lmperm(abschange ~ Centre + Perc_Decomp, data=trt_wide)
## Table of marginal t-test of the betas
## Permutation test using freedman_lane to handle nuisance variables and 5000 permutations.
##                     Estimate Std. Error t value parametric Pr(>|t|)
## (Intercept)          1.67099   0.149332  11.190           3.043e-24
## CentreSingle-centre -0.80570   0.226834  -3.552           4.490e-04
## Perc_Decomp          0.01047   0.002395   4.370           1.756e-05
##                     permutation Pr(<t) permutation Pr(>t) permutation Pr(>|t|)
## (Intercept)                                                                   
## CentreSingle-centre              2e-04              1e+00                4e-04
## Perc_Decomp                      1e+00              2e-04                2e-04
centre_plot <- trt_wide %>% 
  mutate(abschange_dccorr = correct_for_decomp(abschange ~ Centre + Perc_Decomp)) %>% 
  ggplot(aes(x=Centre, y=abschange_dccorr)) +
  geom_beeswarm(aes(colour=Study, group=Study), alpha=0.4, cex=1) +
  guides(colour=FALSE) + 
  geom_smooth(method="lm", se=FALSE) +
  labs(y="Absolute Change (mmHg)\n(after correction for Decompensated Percentage)")

centre_plot

Mean Value

Absolute

permuco::lmperm(abschange ~ meanval + Perc_Decomp, data=trt_wide)
## Table of marginal t-test of the betas
## Permutation test using freedman_lane to handle nuisance variables and 5000 permutations.
##             Estimate Std. Error t value parametric Pr(>|t|) permutation Pr(<t)
## (Intercept) 1.111942   0.417980  2.6603            0.008261                   
## meanval     0.021278   0.024288  0.8761            0.381756             0.8194
## Perc_Decomp 0.006159   0.002201  2.7986            0.005492             0.9974
##             permutation Pr(>t) permutation Pr(>|t|)
## (Intercept)                                        
## meanval                 0.1808               0.3886
## Perc_Decomp             0.0028               0.0056
permTREND(formula=abschange ~ meanval, data=trt_wide_c,
          method="exact.mc")
## 
##  Exact Permutation Test Estimated by Monte Carlo
## 
## data:  x and y
## p-value = 0.396
## alternative hypothesis: true correlation of x and y is not equal to 0
## sample estimates:
## correlation of x and y 
##             0.06545094 
## 
## p-value estimated from 999 Monte Carlo replications
## 99 percent confidence interval on p-value:
##  0.3315721 0.4630919
permTREND(formula=abschange ~ meanval, data=trt_wide_dc,
          method="exact.mc")
## 
##  Exact Permutation Test Estimated by Monte Carlo
## 
## data:  x and y
## p-value = 0.6
## alternative hypothesis: true correlation of x and y is not equal to 0
## sample estimates:
## correlation of x and y 
##              0.0518009 
## 
## p-value estimated from 999 Monte Carlo replications
## 99 percent confidence interval on p-value:
##  0.5250309 0.6760502
meanv_abs_plot <- trt_wide %>% 
  mutate(abschange_dccorr = correct_for_decomp(abschange ~ meanval + Perc_Decomp)) %>% 
  ggplot(aes(x=meanval, y=abschange_dccorr)) +
  geom_point(aes(colour=Study, group=Study), alpha=0.4) +
  guides(colour=FALSE) + 
  geom_smooth(method="lm", se=FALSE) +
  labs(y="Absolute Change (mmHg)\n(after correction for Decompensated Percentage)",
       x="Mean Value across Measurements (mmHg)")

meanv_abs_plot

Signed

meanv_sign <- lmer(change ~ meanval + Perc_Decomp + (1 | Study), data = trt_wide)
summary(meanv_sign)
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: change ~ meanval + Perc_Decomp + (1 | Study)
##    Data: trt_wide
## 
## REML criterion at convergence: 1306.1
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.7727 -0.6196  0.0335  0.5772  3.9536 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Study    (Intercept) 0.5581   0.747   
##  Residual             5.5234   2.350   
## Number of obs: 281, groups:  Study, 21
## 
## Fixed effects:
##               Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)  -2.027119   0.660703  88.680112  -3.068 0.002857 ** 
## meanval       0.124481   0.035183 267.024399   3.538 0.000475 ***
## Perc_Decomp  -0.002642   0.004763  11.767379  -0.555 0.589447    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) meanvl
## meanval     -0.871       
## Perc_Decomp -0.262 -0.104
meanv_sign_nocorr <- lmer(change ~ meanval + (1 | Study), data = trt_wide)
summary(meanv_sign_nocorr)
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: change ~ meanval + (1 | Study)
##    Data: trt_wide
## 
## REML criterion at convergence: 1297.6
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.8493 -0.6062  0.0470  0.5689  3.9289 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Study    (Intercept) 0.505    0.7106  
##  Residual             5.530    2.3515  
## Number of obs: 281, groups:  Study, 21
## 
## Fixed effects:
##             Estimate Std. Error       df t value Pr(>|t|)    
## (Intercept)  -2.1062     0.6338 161.0550  -3.323 0.001102 ** 
## meanval       0.1215     0.0349 258.7515   3.481 0.000586 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##         (Intr)
## meanval -0.938
meanv_sign_c <- lmer(change ~ meanval + (1 | Study), data = trt_wide_c)
summary(meanv_sign_c)
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: change ~ meanval + (1 | Study)
##    Data: trt_wide_c
## 
## REML criterion at convergence: 631.2
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -2.29963 -0.67198  0.00092  0.55771  2.80273 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Study    (Intercept) 0.1048   0.3237  
##  Residual             3.7549   1.9378  
## Number of obs: 150, groups:  Study, 10
## 
## Fixed effects:
##              Estimate Std. Error        df t value Pr(>|t|)   
## (Intercept)  -2.06124    0.71650  94.40528  -2.877  0.00497 **
## meanval       0.12619    0.04241 134.86319   2.976  0.00347 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##         (Intr)
## meanval -0.960
meanv_sign_dc <- lmer(change ~ meanval + (1 | Study), data = trt_wide_dc)
summary(meanv_sign_dc)
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: change ~ meanval + (1 | Study)
##    Data: trt_wide_dc
## 
## REML criterion at convergence: 649.5
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.9742 -0.5787  0.0922  0.6068  3.3725 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Study    (Intercept) 1.057    1.028   
##  Residual             7.583    2.754   
## Number of obs: 131, groups:  Study, 11
## 
## Fixed effects:
##              Estimate Std. Error        df t value Pr(>|t|)  
## (Intercept)  -2.20392    1.06430  88.99352  -2.071   0.0413 *
## meanval       0.12153    0.05574 127.50791   2.180   0.0311 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##         (Intr)
## meanval -0.926
correct_for_decomp_lmer <- function(formula) {
  
  formula <- as.formula(formula)
  
  coefficients <- fixef(lmer(formula, data=trt_wide))

  predicted <- as.character(formula[2])
  
  after_decomp_corr <- trt_wide[[predicted]] - 
    trt_wide[["Perc_Decomp"]] * coefficients[which(names(coefficients)=="Perc_Decomp")]
  
  return(after_decomp_corr)
  
}


meanv_signed_plot <- trt_wide %>% 
  mutate(change_dccorr = correct_for_decomp_lmer("change ~ meanval + Perc_Decomp + (1 | Study)")) %>% 
  ggplot(aes(x=meanval, y=change_dccorr)) +
  geom_jitter(aes(colour=Study, group=Study), alpha=0.4, height = 0.2) +
  guides(colour=FALSE) + 
  geom_smooth(method="lm", se=FALSE) +
  labs(y="Signed Change (mmHg)\n(after correction for Decompensated Percentage)",
       x="Mean Value across Measurements (mmHg)")

meanv_signed_plot

meanv_signed_plot_nocorr <- ggplot(data=trt_wide, aes(x=meanval, y=change)) +
  geom_jitter(aes(colour=Study, group=Study), alpha=0.4, height = 0.2) +
  guides(colour=FALSE) + 
  geom_smooth(method="lm", se=FALSE) +
  labs(y="Signed Change (mmHg)\n(jittered slightly to avoid overlap)",
       x="Mean Value across Measurements (mmHg)")

meanv_signed_plot_nocorr

Figures

cowplot::plot_grid(decomp_plot, days_plot,
                   perc_alc_plot, centre_plot, align = "hv", 
                   ncol = 2, labels = "AUTO")

cowplot::plot_grid(abs_change_distr, sign_change_distr,
                   meanv_abs_plot, meanv_signed_plot, 
                   align = "hv", 
                   ncol = 2, labels = "AUTO")

Power Analysis for a difference

Now, the core thing we want to do here is to perform a power analysis for examining within-individual effects.

One way of doing this is to use the signvar_sd column of the tidytrt object. This is the standard deviation of the signed changes, and hence, if we assume a change after an intervention, this is the SD we could imagine being true, and thus, the effect size, the Cohen’s Dz, is equal to difference / signvar_sd. However, this method makes an assumption that everyone changes by exactly the same amount: the effect (before accounting for error) is completely uniform. This may be the case, but this is the most optimistic scenario. We should be taking into consideration the possibility of heterogeneous effects.

First, let’s make a little plot to show what I mean by homogeneous and heterogeneous effects.

set.seed(1234)

trt_all_comp <- trt_all[2,]

wscv=trt_all_comp$wscv
meanval=trt_all_comp$mean
cv=trt_all_comp$cv
icc=trt_all_comp$icc

sd_true <- sqrt(icc * (cv * meanval)^2)

n <- 20
delta <- 2

# Homogeneous
cv_delta <- 0

pre_true <- rnorm(n, meanval, sd_true)
pre_meas <- pre_true + rnorm(n, 0, meanval*wscv)

post_true <- pre_true - rnorm(n, delta, cv_delta*delta)
post_meas <- post_true + rnorm(n, 0, meanval*wscv)

hom_true <- tibble::tibble(
  ID = rep(1:n, times=2),
  Outcome = c(pre_true, post_true),
  PrePost = rep(c("Pre", "Post"), each=n),
  Effects = "Homogeneous Effects",
  MeasuredTrue = "True Values"
)

hom_measured <- tibble::tibble(
  ID = rep(1:n, times=2),
  Outcome = c(pre_meas, post_meas),
  PrePost = rep(c("Pre", "Post"), each=n),
  Effects = "Homogeneous Effects",
  MeasuredTrue = "Measured Values"
)

# hom_difference <- tibble::tibble(
#   ID = rep(1:n, times=2),
#   Outcome = c(post_meas-pre_),
#   PrePost = rep(c("Pre", "Post"), each=n),
#   Effects = "Homogeneous",
#   MeasuredTrue = "Difference"
# )


# Heterogeneous
cv_delta <- 0.5

#pre_true <- rnorm(n, meanval, abs(cv*meanval)) # Use same as above
#pre_meas <- pre_true + rnorm(n, 0, abs(meanval*wscv))

post_true <- pre_true - rnorm(n, delta, abs(cv_delta*delta))
post_meas <- post_true + rnorm(n, 0, abs(meanval*wscv))

het_true <- tibble::tibble(
  ID = rep(1:n, times=2),
  Outcome = c(pre_true, post_true),
  PrePost = rep(c("Pre", "Post"), each=n),
  Effects = "Heterogeneous Effects",
  MeasuredTrue = "True Values"
)

het_measured <- tibble::tibble(
  ID = rep(1:n, times=2),
  Outcome = c(pre_meas, post_meas),
  PrePost = rep(c("Pre", "Post"), each=n),
  Effects = "Heterogeneous Effects",
  MeasuredTrue = "Measured Values"
)

# Plot
effects <- bind_rows(hom_true, hom_measured, het_true, het_measured) %>% 
  mutate(MeasuredTrue = fct_inorder(MeasuredTrue),
         Effects = fct_inorder(Effects),
         PrePost = fct_inorder(PrePost),
         ID = as.factor(ID))

ggplot(effects, aes(x=PrePost, y=Outcome, colour=ID, group=ID)) +
  geom_point(size=2) +
  geom_line() +
  facet_grid(Effects~MeasuredTrue) +
  labs(y="Outcome (mmHg)",
       colour="Values",
       x=NULL,
       title="Homogeneous and Heteregeneous Effects",
       subtitle="Homogeneous effects imply that the true underlying change is the same\nin all individuals (hence parallel lines in the true values)") +
  guides(colour=FALSE)

So, to summarise, we have underlying true values, and measured values after accounting for measurement error. The change from before to after the intervention can either be homogeneous (everyone has exactly the same effect), or heterogeneous (effect sizes differ, and some even get harmed by the intervention - about 2.5% as I’ve chosen the SD of the intervention effect as 50% of the mean effect, so 0 effect is 2 SDs away from the mean effect size, which is approximately 2.5%). Then, the measured values appear to show more people getting worse after treatment, but this is just due to measurement error.

heterogen_cv <- 0.5

annotations <- tibble(
  x=c(-3, 0),
  text=c(paste0(round(100*pnorm(1/heterogen_cv)), "% experience improvements"),
         paste0(100-round(100*pnorm(1/heterogen_cv)), "% experience worsening")),
  colour = c("#61b096", "#bd7969")
)

heterogen_effects <- tibble(Effect=c(-3, 1)) 

ggplot(heterogen_effects, aes(x=Effect)) +
  geom_area(stat="function", fun = dnorm, fill="#61b096", xlim=c(-3, 0),
            args = list(mean = -1, sd=heterogen_cv), alpha=0.7) +
  geom_area(stat="function", fun = dnorm, fill="#bd7969", xlim=c(0, 1),
            args = list(mean = -1, sd=heterogen_cv), alpha=0.7) +
  annotate("text", x=-2.5, y=0.7, 
           label=paste0(round(100*pnorm(1/heterogen_cv), 1), 
                        "% experience\nimprovements"),
           colour = "#61b096", hjust=0.5) +
  annotate("text", x=0.5, y=0.7, 
           label=paste0(100-round(100*pnorm(1/heterogen_cv), 1), 
                        "% experience\nworsening"),
           colour = "#bd7969", hjust=0.5) +
  annotate("text", x=-0.8, y=0.2, 
           label="Mean\neffect",hjust=0.5) +
  theme(axis.title.y=element_blank(),
        axis.text.y=element_blank(),
        axis.ticks.y=element_blank()) +
  geom_vline(xintercept = -1) +
  labs(title="Heterogeneous Effects",
       subtitle=paste0("Using a 50% CV for effect heterogeneity implies that some ",
                       "participants\nmay benefit more and others may even have ",
                       "worsening."))

This would imply that for all different sizes of effect, that only 2.3% experience a true worsening. However, when we measure the values, there are also several individuals who will exhibit an apparent worsening (increases from first to second measurement), when their true underlying values exhibited improvements. Let’s calculate what fraction of individuals this would be.

True and Apparent Changes

Here, we can also observe the percentage of individuals who show apparent 10% and 20% changes from baseline.

apparent_effects <- function(n, delta, cv_delta, wscv=trt_all$wscv, 
                     mean=trt_all$mean, cv=trt_all$cv, icc=trt_all$icc, 
                     decomp) {
  
  wscv <- wscv[decomp]
  mean <- mean[decomp]
  cv   <- cv[decomp]
  
  var <- (cv*mean)^2
  sd_true <- sqrt(var * icc)
  
  
  pre_true <- rnorm(n, mean, sd_true)
  
  pre_meas <- pre_true + rnorm(n, 0, mean*wscv)
  
  post_true <- pre_true - rnorm(n, delta, cv_delta*delta)
  
  post_meas <- post_true + rnorm(n, 0, mean*wscv)
  
  measured <- tibble::tibble(
    ID = rep(1:n, times=2),
    Outcome = c(pre_meas, post_meas),
    PrePost = rep(c("Pre", "Post"), each=n)
  ) %>% 
    spread(PrePost, Outcome)
  
  true <- tibble::tibble(
    ID = rep(1:n, times=2),
    Outcome = c(pre_true, post_true),
    PrePost = rep(c("Pre", "Post"), each=n)
  ) %>% 
    spread(PrePost, Outcome)
  
  out <- list()
  
  out$apparent_worse <- round(100*with(measured, mean(Post > Pre)),1)
  out$apparent_10    <- round(100*with(measured, mean((Pre-Post)/Pre > 0.1)),1)
  out$apparent_20    <- round(100*with(measured, mean((Pre-Post)/Pre > 0.2)),1)
  out$true_worse     <- round(100*with(true, mean(Post > Pre)),1)
  out$true_10        <- round(100*with(true, mean((Pre-Post)/Pre > 0.1)),1)
  out$true_20        <- round(100*with(true, mean((Pre-Post)/Pre > 0.2)),1)
  
  return(out)

}

if(!file.exists("../DerivedData/percdifs.rds") || overwrite) {

measured_percs <- tidyr::crossing(
  delta = seq(0, 3, by=0.5),
  cv_delta = c(0, 0.5),
  decomp=c(1,2)
) %>%
  mutate(condition = 1:n()) %>%
  group_by(condition) %>%
  nest() %>%
  mutate(res = map(data, ~apparent_effects( n=1e7,
                                               delta=.x$delta,
                                               cv_delta = .x$cv_delta,
                                               decomp=.x$decomp))) %>%
  ungroup()

saveRDS(measured_percs, "../DerivedData/percdifs.rds")

}
measured_percs <- readRDS("../DerivedData/percdifs.rds")

measured_percs_summary <- measured_percs %>% 
  mutate(apparent_worse = map_dbl(res, "apparent_worse"),
         apparent_10 = map_dbl(res, "apparent_10"),
         apparent_20 = map_dbl(res, "apparent_20"),
         true_worse = map_dbl(res, "true_worse"),
         true_10 = map_dbl(res, "true_10"),
         true_20 = map_dbl(res, "true_20")) %>% 
  select(-res) %>% 
  unnest(data) %>% 
  #mutate(true_worse = round(100*(1-pnorm(1/cv_delta)),1)) %>% 
  arrange(decomp, delta, cv_delta) %>% 
  mutate(decomp = ifelse(decomp==1, 
                         trt_all$decomp[1], 
                         trt_all$decomp[2]),
         cv_delta = ifelse(cv_delta==0,
                           "Homogeneous",
                           "Heterogeneous")) %>% 
  select(decomp, condition, delta, cv_delta,
         true_worse, apparent_worse,
         true_10, apparent_10, 
         true_20, apparent_20) %>% 
  rename("Patients" = decomp,
         "Apparent Worsening (%)" = apparent_worse,
         "True Worsening (%)" = true_worse,
         "True 10%+ Improvement (%)" = true_10,
         "Apparent 10%+ Improvement (%)" = apparent_10,
         "True 20%+ Improvement (%)" = true_20,
         "Apparent 20%+ Improvement (%)" = apparent_20,
         "Effects" = cv_delta,
         "Difference" = delta) %>% 
  select(-condition)

decomp_change <- head(
  which(
    measured_percs_summary$Patients ==
      trt_all$decomp[2]), 1)

knitr::kable(measured_percs_summary[,-1]) %>% 
  kable_styling("striped", full_width = F) %>%
  pack_rows(trt_all$decomp[1], 1, decomp_change-1) %>%
  pack_rows(trt_all$decomp[2], decomp_change, nrow(measured_percs_summary))
Difference Effects True Worsening (%) Apparent Worsening (%) True 10%+ Improvement (%) Apparent 10%+ Improvement (%) True 20%+ Improvement (%) Apparent 20%+ Improvement (%)
Includes Decompensated
0.0 Homogeneous 0.0 50.0 0.0 25.4 0.0 9.1
0.0 Heterogeneous 0.0 50.0 0.0 25.4 0.0 9.1
0.5 Homogeneous 0.0 42.8 0.2 31.8 0.0 12.7
0.5 Heterogeneous 2.3 42.8 0.6 31.9 0.0 12.8
1.0 Homogeneous 0.0 35.8 3.9 38.8 0.2 17.1
1.0 Heterogeneous 2.3 36.0 12.7 39.0 0.6 17.5
1.5 Homogeneous 0.0 29.2 27.8 46.3 1.0 22.3
1.5 Heterogeneous 2.3 29.9 38.6 46.3 3.9 23.2
2.0 Homogeneous 0.0 23.3 72.0 53.8 3.9 28.4
2.0 Heterogeneous 2.3 24.7 59.1 53.5 12.7 29.7
2.5 Homogeneous 0.0 18.1 96.0 61.2 12.0 35.1
2.5 Heterogeneous 2.3 20.4 71.5 60.1 25.4 36.5
3.0 Homogeneous 0.0 13.7 99.8 68.2 27.9 42.4
3.0 Heterogeneous 2.3 16.9 78.8 65.9 38.6 43.4
Only Compensated
0.0 Homogeneous 0.0 50.0 0.0 17.7 0.0 3.5
0.0 Heterogeneous 0.0 50.0 0.0 17.7 0.0 3.5
0.5 Homogeneous 0.0 38.7 0.2 26.3 0.0 6.4
0.5 Heterogeneous 2.3 38.9 0.7 26.5 0.0 6.6
1.0 Homogeneous 0.0 28.3 5.7 36.7 0.2 10.8
1.0 Heterogeneous 2.3 29.1 17.6 37.2 0.7 11.7
1.5 Homogeneous 0.0 19.5 41.7 48.2 1.2 17.1
1.5 Heterogeneous 2.3 21.5 46.3 48.4 5.7 19.2
2.0 Homogeneous 0.0 12.6 87.7 59.9 5.7 25.4
2.0 Heterogeneous 2.3 16.0 65.4 58.5 17.6 28.3
2.5 Homogeneous 0.0 7.6 99.4 70.7 18.5 35.4
2.5 Heterogeneous 2.3 12.2 76.1 66.9 32.6 38.0
3.0 Homogeneous 0.0 4.3 100.0 79.9 41.7 46.5
3.0 Heterogeneous 2.3 9.6 82.2 73.5 46.3 47.3

So, for a true effect of about 2mmHg in compensated patients, it will appear as if 12-16% would appear to worsen. This fits with clinical experience.

Simulation

HVPG_dif_sim <- function(n, delta, cv_delta, wscv=trt_all$wscv, 
                     mean=trt_all$mean, cv=trt_all$cv, icc=trt_all$icc,
                     decomp = 1) {
  
  wscv <- wscv[decomp]
  mean <- mean[decomp]
  cv   <- cv[decomp]
  
  var <- (cv*mean)^2
  sd_true <- sqrt(var * icc)
  
  pre_true <- rnorm(n, mean, sd_true)
  
  pre_meas <- pre_true + rnorm(n, 0, mean*wscv)
  
  post_true <- pre_true - rnorm(n, delta, cv_delta*delta)
  
  post_meas <- post_true + rnorm(n, 0, mean*wscv)
  
  measured <- tibble::tibble(
    ID = rep(1:n, times=2),
    Outcome = c(pre_meas, post_meas),
    PrePost = rep(c("Pre", "Post"), each=n)
  )
  
  d <- effsize::cohen.d(measured$Outcome, measured$PrePost, 
                        paired=TRUE)$estimate
  
  dz <- effsize::cohen.d(measured$Outcome, measured$PrePost, 
                        paired=TRUE, within=FALSE)$estimate
  
  test <- t.test(pre_meas, post_meas, alternative = "greater", 
              paired = T)
  
  # Note: one-sided p value
  
  testout <- broom::tidy(test)
  
  out <- mutate(testout, d = d, dz = dz)
  
  return(out)
  
}

Now, we set up the simulation parameters for various scenarios.

difsimpars <- tidyr::crossing(
  n=seq(5, 100, by = 5),
  delta = seq(1,3, by=0.5),
  cv_delta = c(0, 0.5),
)

And now we run them

if(!file.exists(paste0("../DerivedData/difsims_decomp_", 
                       nsims, ".rds")) || overwrite) {
  
  pb <- progress_bar$new(total = nrow(difsimpars))
  
  difsims <- difsimpars %>% 
    mutate(sim = 1:nrow(difsimpars)) %>% 
    group_by(sim) %>% 
    nest(params = c(n, delta, cv_delta)) %>% 
    mutate(output = map(params,
                        ~{pb$tick(); 
                          bind_rows(purrr::rerun(nsims, 
                            HVPG_dif_sim(.x$n, .x$delta, 
                                       .x$cv_delta, 
                                       decomp=1)))}))

  saveRDS(difsims, paste0("../DerivedData/difsims_decomp_", nsims, ".rds"))
  
}
if(!file.exists(paste0("../DerivedData/difsims_comp_", 
                       nsims, ".rds")) || overwrite) {
  
  pb <- progress_bar$new(total = nrow(difsimpars))
  
  difsims <- difsimpars %>% 
    mutate(sim = 1:nrow(difsimpars)) %>% 
    group_by(sim) %>% 
    nest(params = c(n, delta, cv_delta)) %>% 
    mutate(output = map(params,
                        ~{pb$tick(); 
                          bind_rows(purrr::rerun(nsims, 
                            HVPG_dif_sim(.x$n, .x$delta, 
                                       .x$cv_delta,  
                                       decomp=2)))}))

  saveRDS(difsims, paste0("../DerivedData/difsims_comp_", nsims, ".rds"))
  
}

And extract the results

difsims_decomp <- readRDS(
  paste0("../DerivedData/difsims_decomp_", nsims, ".rds"))

difsims_decomp_res <- difsims_decomp %>% 
  ungroup() %>% 
  mutate(power = map_dbl(output, ~mean(.x$p.value < 0.05))) %>% 
  unnest(params) %>% 
  mutate(delta = as.factor(delta)) %>% 
  mutate(Effects = ifelse(cv_delta==0, "Homogeneous Effects", 
                                       "Heterogeneous Effects")) %>% 
  mutate(Effects = fct_inorder(Effects)) %>% 
  mutate(decomp = trt_all$decomp[1])





difsims_comp <- readRDS(
  paste0("../DerivedData/difsims_comp_", nsims, ".rds"))

difsims_comp_res <- difsims_comp %>% 
  ungroup() %>% 
  mutate(power = map_dbl(output, ~mean(.x$p.value < 0.05))) %>% 
  unnest(params) %>% 
  mutate(delta = as.factor(delta)) %>% 
  mutate(Effects = ifelse(cv_delta==0, "Homogeneous Effects", 
                                       "Heterogeneous Effects")) %>% 
  mutate(Effects = fct_inorder(Effects)) %>% 
  mutate(decomp = trt_all$decomp[2])




difsims_res <- bind_rows(difsims_comp_res, difsims_decomp_res)

Plotting

ggplot(difsims_res, aes(x=n, y=power, colour=delta)) +
  geom_point() + 
  geom_line() +
  facet_grid(decomp~Effects) +
  coord_cartesian(ylim=c(0.5, 1)) +
  scale_color_brewer(type = "qual", palette = 2) +
  annotate("rect", ymin = 0.8, ymax = 1, xmin=0, 
           xmax=100, alpha = .4, fill="grey") +
  labs(y="Power", x="Sample Size", 
       colour="Intervention\nEffect (mmHg)")

Required Individuals

And how many people do we need for each scenario?

difsims_80power <- difsims_res %>% 
  arrange(power) %>% 
  filter(power > 0.8) %>% 
  select(decomp, delta, Effects, n) %>% 
  group_by(delta, Effects, decomp) %>% 
  slice(1) %>% 
  arrange(decomp, delta) %>% 
  rename("Patients" = decomp,
         "Difference (mmHg)" = delta,
         "80% Power" = n)

difsims_90power <- difsims_res %>% 
  arrange(power) %>% 
  filter(power > 0.9) %>% 
  select(decomp, delta, Effects, n) %>% 
  group_by(delta, Effects, decomp) %>% 
  slice(1) %>% 
  arrange(decomp, delta) %>% 
  rename("Patients" = decomp,
         "Difference (mmHg)" = delta,
         "90% Power" = n)

difsims_power <- left_join(difsims_80power, difsims_90power)

decomp_change <- head(
  which(
    difsims_power$Patients ==
      trt_all$decomp[2]), 1)

kable(difsims_power[,-1]) %>% 
  kable_styling("striped", full_width = F) %>%
  pack_rows(trt_all$decomp[1], 1, decomp_change-1) %>%
  pack_rows(trt_all$decomp[2], decomp_change, nrow(difsims_80power))
Difference (mmHg) Effects 80% Power 90% Power
Includes Decompensated
1 Homogeneous Effects 55 75
1 Heterogeneous Effects 60 80
1.5 Homogeneous Effects 25 35
1.5 Heterogeneous Effects 30 40
2 Homogeneous Effects 15 20
2 Heterogeneous Effects 20 25
2.5 Homogeneous Effects 10 15
2.5 Heterogeneous Effects 15 20
3 Homogeneous Effects 10 10
3 Heterogeneous Effects 10 15
Only Compensated
1 Homogeneous Effects 30 40
1 Heterogeneous Effects 30 40
1.5 Homogeneous Effects 15 20
1.5 Heterogeneous Effects 15 20
2 Homogeneous Effects 10 15
2 Heterogeneous Effects 10 15
2.5 Homogeneous Effects 10 10
2.5 Heterogeneous Effects 10 10
3 Homogeneous Effects 5 10
3 Heterogeneous Effects 10 10

Analytical solution

Let’s just check these with the analytical solution

  • Decomp

  • Homogeneous

  • 1.5

  • Expect: 20 < n < 25

library(pwr)

dz <- 1.5/(trt_all$signvar_sd[1]*trt_all$mean[1])

pwr::pwr.t.test(d=dz, sig.level = 0.05, power = 0.8, 
                type = "paired", alternative = "greater")
## 
##      Paired t test power calculation 
## 
##               n = 24.65095
##               d = 0.5157783
##       sig.level = 0.05
##           power = 0.8
##     alternative = greater
## 
## NOTE: n is number of *pairs*

And hetero

  • Decomp

  • Homogeneous

  • 1.5

  • Expect: 25 < n < 30

signvar_sd <- sqrt(
    (trt_all$signvar_sd[1]*trt_all$mean[1])^2 +
    (0.5*1.5)^2)
  

dz <- 1.5/signvar_sd

pwr::pwr.t.test(d=dz, sig.level = 0.05, power = 0.8, 
                type = "paired", alternative = "greater")
## 
##      Paired t test power calculation 
## 
##               n = 26.19313
##               d = 0.4994376
##       sig.level = 0.05
##           power = 0.8
##     alternative = greater
## 
## NOTE: n is number of *pairs*
difsims_ana <- function(Patients, Difference, Effects, Power,
                        trt_all=trt_all) {
  
  Effects <- as.character(Effects)
  Difference <- as.numeric(Difference)
  
  heterogen <- ifelse(Effects=="Homogeneous Effects",
                      0, 0.5)
  
  decomp = ifelse(Patients=="Includes Decompensated", 1, 2)
  
  trt_all_pat <- trt_all[decomp,]
  
  signvar_sd <- sqrt(
    (trt_all_pat$signvar_sd*trt_all_pat$mean)^2 +
    (heterogen*Difference)^2)
  
  dz <- Difference/signvar_sd
  
  ceiling(pwr::pwr.t.test(d=dz, sig.level = 0.05, power = Power,
                       alternative = "greater", type = "paired")$n)
  
}

difsims_power_ana <- difsims_power %>% 
  rename(Difference = `Difference (mmHg)`) %>% 
  mutate(Difference = as.numeric(as.character(Difference)),
         Effects= as.character(Effects)) %>% 
  group_by(Patients, Difference, Effects) %>% 
  mutate("80% Power"= pmap_dbl(list(Patients, Difference,
                                    Effects), difsims_ana, Power=0.8,
                                   trt_all = trt_all),
         "90% Power"= pmap_dbl(list(Patients, Difference,
                                    Effects), difsims_ana, Power=0.9,
                                   trt_all = trt_all))

kable(difsims_power_ana[,-1]) %>% 
  kable_styling("striped", full_width = F) %>%
  pack_rows(trt_all$decomp[1], 1, decomp_change-1) %>%
  pack_rows(trt_all$decomp[2], decomp_change, nrow(difsims_power_ana))
Difference Effects 80% Power 90% Power
Includes Decompensated
1.0 Homogeneous Effects 54 74
1.0 Heterogeneous Effects 56 76
1.5 Homogeneous Effects 25 34
1.5 Heterogeneous Effects 27 36
2.0 Homogeneous Effects 15 20
2.0 Heterogeneous Effects 17 22
2.5 Homogeneous Effects 10 14
2.5 Heterogeneous Effects 12 16
3.0 Homogeneous Effects 8 10
3.0 Heterogeneous Effects 9 12
Only Compensated
1.0 Homogeneous Effects 27 36
1.0 Heterogeneous Effects 28 39
1.5 Homogeneous Effects 13 17
1.5 Heterogeneous Effects 15 19
2.0 Homogeneous Effects 8 11
2.0 Heterogeneous Effects 10 13
2.5 Homogeneous Effects 6 8
2.5 Heterogeneous Effects 8 10
3.0 Homogeneous Effects 5 6
3.0 Heterogeneous Effects 6 8

Contour Plots

difsims_ana_contour <- function(Patients, Difference, Effects, n,
                        trt_all=trt_all) {
  
  Effects <- as.character(Effects)
  Difference <- as.numeric(Difference)
  
  heterogen <- ifelse(Effects=="Homogeneous Effects",
                      0, 0.5)
  
  decomp = ifelse(Patients=="Includes Decompensated", 1, 2)
  
  trt_all_pat <- trt_all[decomp,]
  
  signvar_sd <- sqrt(
    (trt_all_pat$signvar_sd*trt_all_pat$mean)^2 +
    (heterogen*Difference)^2)
  
  dz <- Difference/signvar_sd
  
  pwr::pwr.t.test(d=dz, sig.level = 0.05, n = n,
                       alternative = "greater", type = "paired")$power
  
}

contour_dat <- tidyr::crossing(Difference = seq(0.3,3,length.out=2701),
                               n = 5:80,
                               Effects = c("Homogeneous Effects", 
                                           "Heterogeneous Effects"),
                               Patients = c("Includes Decompensated", 
                                            "Only Compensated"))

Run it

contour_power <- contour_dat %>% 
  mutate(Test = 1:n()) %>% 
  group_by(Test) %>% 
  mutate(Power=pmap_dbl(list(Patients, Difference, Effects, n), 
                        difsims_ana_contour, trt_all=trt_all)) %>% 
  ungroup()

saveRDS(contour_power, "../DerivedData/contour_power.rds")
contour_power <- readRDS("../DerivedData/contour_power.rds")

library(viridis)

contour_power <- contour_power %>% 
  mutate(Power_cut = cut(Power, breaks = seq(0,1, length.out = 11))) %>% 
  mutate(Power = ifelse(Power == 1, 0.99, Power)) # Note below to explain this
                                                  

contour_het <- contour_power %>% 
  filter(Effects!="Homogeneous Effects")

contour_hom <- contour_power %>% 
  filter(Effects=="Homogeneous Effects")


homplot <- ggplot(contour_hom, aes(x=n, y=Difference, z=Power)) +
  geom_contour_filled(alpha=0.8, breaks=seq(0,1, by=0.1)) +
  facet_wrap(.~Patients, scales = "free") +
  theme_ipsum_rc() +
  labs(x = "Sample Size",
       y = "Hypothetical True Intervention Effect (mmHg)") +
  scale_y_continuous(breaks = seq(0.5, 3, by = 0.5)) +
  scale_fill_viridis("Power", discrete = T) +
  geom_contour(breaks=0.8, colour="black", linetype="dashed")


hetplot <- ggplot(contour_het, aes(x=n, y=Difference, z=Power)) +
  geom_contour_filled(alpha=0.8, breaks=seq(0,1.1, by=0.1)) +
  facet_wrap(.~Patients, scales = "free") +
  theme_ipsum_rc() +
  labs(x = "Sample Size",
       y = "Hypothetical True Intervention Effect (mmHg)") +
  scale_y_continuous(breaks = seq(0.5, 3, by = 0.5)) +
  scale_fill_viridis("Power", discrete = T) +
  geom_contour(breaks=0.8, colour="black", linetype="dashed")

homplot

hetplot

ggsave(homplot, height=5, width=10, filename = "figures/Dif_hom_contour.png")
ggsave(hetplot, height=5, width=10, filename = "figures/Dif_het_contour.png")

ggsave(homplot, height=5, width=10, filename = "figures/Dif_hom_contour.jpg", 
       dpi = 600)
ggsave(hetplot, height=5, width=10, filename = "figures/Dif_het_contour.jpg", 
       dpi = 600)

# +
#   xlab("Minor allele frequency")+
#   ylab("Hypothetical effect size")+
#   scale_x_log10(expand = c(0, 0), position = "bottom") +
#   scale_y_continuous(expand = c(0, 0))+
#   geom_line(data = power.80, col = "black")+
#   theme_classic(base_size = 12)

The strange line of code where I convert those values = 1 to 0.99 is to prevent the graph from showing strangely. I think this has to do with floating point accuracy. Some of the values equal to 1, are rounded in the computer number system to a little bit above 1, and then the plot makes them white. So by setting them to 0.99, they’re still the same colour, but the plot is filled correctly.

Power Analysis for Difference in Differences

Simulation

HVPG_difindif_sim <- function(n, delta1, delta2, cv_delta, 
                              wscv=trt_all$wscv, 
                              mean=trt_all$mean, 
                              cv=trt_all$cv, 
                              decomp ) {
  
  wscv <- wscv[decomp]
  mean <- mean[decomp]
  cv   <- cv[decomp]
  
  var <- (cv*mean)^2
  sd_true <- sqrt(var * icc)
  
  pre_true1 <- rnorm(n, mean, sd_true)
  pre_true2 <- rnorm(n, mean, sd_true)
  
  pre_meas1 <- pre_true1 + rnorm(n, 0, mean*wscv)
  pre_meas2 <- pre_true2 + rnorm(n, 0, mean*wscv)
  
  post_true1 <- pre_true1 - rnorm(n, delta1, cv_delta*delta1)
  post_true2 <- pre_true2 - rnorm(n, delta2, cv_delta*delta2)
  
  post_meas1 <- post_true1 + rnorm(n, 0, mean*wscv)
  post_meas2 <- post_true2 + rnorm(n, 0, mean*wscv)
  
  measured <- tibble::tibble(
    ID = rep(1:n, times=2),
    Pre = c(pre_meas1, pre_meas2),
    Post = c(post_meas1, post_meas2),
    Diff = Post - Pre,
    Group = rep(c("A", "B"), each=n)
  )
  
  d <- effsize::cohen.d(measured$Diff, measured$Group, 
                        paired=FALSE)$estimate
  
  mod <- lm(Post ~ Pre + Group, data=measured)
  
  testout <- broom::tidy(mod) %>% 
    filter(term=="GroupB") %>% 
    select(-term) %>% 
    mutate(p.value = pt(statistic, mod$df, lower.tail = FALSE))
  # Note: using a one-sided p value
  
  out <- mutate(testout, d=d)
  
  return(out)
  
}

Now, we set up the simulation for various scenarios.

difindifsimpars <- tidyr::crossing(
  n = c( seq(5, 100, by = 5), seq(110, 200, by=10)),
  delta1 = seq(1,3, by = 0.5),
  delta2 = seq(0,2, by = 0.5),
  cv_delta = c(0, 0.5),
) %>% 
  mutate(deltadif = delta1 - delta2) %>% 
  filter(delta1 > delta2)

And we run it

if(!file.exists(paste0("../DerivedData/difindifsims_decomp_", 
                       nsims, ".rds")) || overwrite) {
  
  pb <- progress_bar$new(total = nrow(difindifsimpars))
  
  difindifsims <- difindifsimpars %>% 
    mutate(sim = 1:nrow(difindifsimpars)) %>% 
    group_by(sim) %>% 
    nest(params = c(n, delta1, delta2, cv_delta)) %>% 
    mutate(output = map(params, .progress = TRUE,
                        ~{pb$tick(); 
                          bind_rows(purrr::rerun(nsims, 
                          HVPG_difindif_sim(.x$n, .x$delta1, 
                                            .x$delta2, .x$cv_delta, 
                                            decomp = 1)))}))
  
  saveRDS(difindifsims, 
          paste0("../DerivedData/difindifsims_decomp_", nsims, ".rds"))
  
}
if(!file.exists(paste0("../DerivedData/difindifsims_comp_", 
                       nsims, ".rds")) || overwrite) {
  
  pb <- progress_bar$new(total = nrow(difindifsimpars))
  
  difindifsims <- difindifsimpars %>% 
    mutate(sim = 1:nrow(difindifsimpars)) %>% 
    group_by(sim) %>% 
    nest(params = c(n, delta1, delta2, cv_delta)) %>% 
    mutate(output = map(params, 
                        ~{pb$tick(); 
                          bind_rows(purrr::rerun(nsims, 
                          HVPG_difindif_sim(.x$n, .x$delta1, 
                                            .x$delta2, .x$cv_delta,
                                            decomp = 2)))}))
  
  saveRDS(difindifsims, 
          paste0("../DerivedData/difindifsims_comp_", nsims, ".rds"))
  
}

And extract the results

difindifsims_decomp <- readRDS(paste0("../DerivedData/difindifsims_decomp_", nsims, ".rds"))

difindifsims_decomp_res <- difindifsims_decomp %>% 
  ungroup() %>% 
  mutate(power = map_dbl(output, ~mean(.x$p.value < 0.05))) %>% 
  unnest(params) %>% 
  mutate(deltadif = delta1 - delta2,
         delta1 = as.factor(delta1),
         delta2 = as.factor(delta2),
         deltadif = as.factor(deltadif)) %>% 
  #filter(deltadif != 0.5) %>% 
  mutate(Effects = ifelse(cv_delta==0, "Homogeneous Effects", 
                                       "Heterogeneous Effects")) %>% 
  mutate(Effects = fct_inorder(Effects)) %>% 
  mutate(decomp = trt_all$decomp[1])


difindifsims_comp <- readRDS(paste0("../DerivedData/difindifsims_comp_", nsims, ".rds"))

difindifsims_comp_res <- difindifsims_comp %>% 
  ungroup() %>% 
  mutate(power = map_dbl(output, ~mean(.x$p.value < 0.05))) %>% 
  unnest(params) %>% 
  mutate(deltadif = delta1 - delta2,
         delta1 = as.factor(delta1),
         delta2 = as.factor(delta2),
         deltadif = as.factor(deltadif)) %>% 
  #filter(deltadif != 0.5) %>% 
  mutate(Effects = ifelse(cv_delta==0, "Homogeneous Effects", 
                                       "Heterogeneous Effects")) %>% 
  mutate(Effects = fct_inorder(Effects)) %>% 
  mutate(decomp = trt_all$decomp[2])



difindifsims_res <- bind_rows(difindifsims_decomp_res, 
                              difindifsims_comp_res)

Plotting

Here we have the size of the effect of the better intervention as the

difindifplot_decomp <-
  ggplot(difindifsims_decomp_res, aes(x=n, y=power, colour=delta2)) +
    geom_point() + 
    geom_line() +
    facet_grid(delta1~Effects) +
    coord_cartesian(ylim=c(0.5, 1)) +
    annotate("rect", ymin = 0.8, ymax = 1, xmin=0, xmax=200, 
             alpha = .4, fill="grey") +
    scale_color_brewer(type = "qual", palette = 2) +
    labs(y="Power", x="Sample Size", 
       colour="Reference\nIntervention\nEffect (mmHg)",
       title=trt_all$decomp[1]) +
    theme(plot.title = element_text(hjust = 0.5))

difindif_legend <- cowplot::get_legend(difindifplot_decomp)

difindifplot_decomp <- difindifplot_decomp +
  guides(colour=FALSE)

difindifplot_comp <-
  ggplot(difindifsims_comp_res, aes(x=n, y=power, colour=delta2)) +
    geom_point() + 
    geom_line() +
    facet_grid(delta1~Effects) +
    coord_cartesian(ylim=c(0.5, 1)) +
    annotate("rect", ymin = 0.8, ymax = 1, xmin=0, xmax=200, 
             alpha = .4, fill="grey") +
    scale_color_brewer(type = "qual", palette = 2) +
    labs(y="Power", x="Sample Size", 
       colour="Reference\nIntervention\nEffect (mmHg)",
       title=trt_all$decomp[2]) +
    theme(plot.title = element_text(hjust = 0.5)) +
    guides(colour=FALSE)

difindifplot <- cowplot::plot_grid(
  difindifplot_decomp, difindifplot_comp, difindif_legend,
  nrow = 1, rel_widths = c(3,3,0.5)
)

difindifplot

Required Individuals

And how many people do we need for each scenario?

difindifsims_80power <- difindifsims_res %>% 
  arrange(power) %>% 
  filter(power > 0.8) %>% 
  select(decomp, deltadif, delta1, delta2, Effects, n) %>% 
  group_by(delta1, deltadif, Effects, decomp) %>% 
  slice(1) %>% 
  arrange(decomp, desc(deltadif), desc(delta1), Effects) %>% 
  rename("Patients" = decomp,
         "Intervention 1 Effect (mmHg)" = delta1,
         "Intervention 2 Effect (mmHg)" = delta2,
         "Difference (mmHg)" = deltadif,
         "80% Power" = n)

difindifsims_90power <- difindifsims_res %>% 
  arrange(power) %>% 
  filter(power > 0.9) %>% 
  select(decomp, deltadif, delta1, delta2, Effects, n) %>% 
  group_by(delta1, deltadif, Effects, decomp) %>% 
  slice(1) %>% 
  arrange(decomp, desc(deltadif), desc(delta1), Effects) %>% 
  rename("Patients" = decomp,
         "Intervention 1 Effect (mmHg)" = delta1,
         "Intervention 2 Effect (mmHg)" = delta2,
         "Difference (mmHg)" = deltadif,
         "90% Power" = n)

difindifsims_power <- left_join(difindifsims_80power, difindifsims_90power) %>%
  mutate(`90% Power` = as.character(`90% Power`)) %>% 
  mutate(`90% Power` = ifelse(is.na(`90% Power`),
                              ">200", `90% Power`)) %>% 
  filter(`Difference (mmHg)` != 0.5)

decomp_change <- head(
  which(
    difindifsims_power$Patients ==
      trt_all$decomp[2]), 1)

kable(difindifsims_power[,-1]) %>% 
  kable_styling("striped", full_width = F) %>%
  pack_rows(trt_all$decomp[1], 1, decomp_change-1) %>%
  pack_rows(trt_all$decomp[2], decomp_change, nrow(difindifsims_power))
Difference (mmHg) Intervention 1 Effect (mmHg) Intervention 2 Effect (mmHg) Effects 80% Power 90% Power
Includes Decompensated
3 3 0 Homogeneous Effects 15 20
3 3 0 Heterogeneous Effects 15 20
2.5 3 0.5 Homogeneous Effects 20 25
2.5 3 0.5 Heterogeneous Effects 20 30
2.5 2.5 0 Homogeneous Effects 20 25
2.5 2.5 0 Heterogeneous Effects 20 25
2 3 1 Homogeneous Effects 30 35
2 3 1 Heterogeneous Effects 30 40
2 2.5 0.5 Homogeneous Effects 25 35
2 2.5 0.5 Heterogeneous Effects 30 40
2 2 0 Homogeneous Effects 30 35
2 2 0 Heterogeneous Effects 30 40
1.5 3 1.5 Homogeneous Effects 45 65
1.5 3 1.5 Heterogeneous Effects 55 75
1.5 2.5 1 Homogeneous Effects 45 60
1.5 2.5 1 Heterogeneous Effects 50 70
1.5 2 0.5 Homogeneous Effects 45 60
1.5 2 0.5 Heterogeneous Effects 50 65
1.5 1.5 0 Homogeneous Effects 45 60
1.5 1.5 0 Heterogeneous Effects 50 60
1 3 2 Homogeneous Effects 100 140
1 3 2 Heterogeneous Effects 120 170
1 2.5 1.5 Homogeneous Effects 100 140
1 2.5 1.5 Heterogeneous Effects 110 150
1 2 1 Homogeneous Effects 100 140
1 2 1 Heterogeneous Effects 110 150
1 1.5 0.5 Homogeneous Effects 100 140
1 1.5 0.5 Heterogeneous Effects 110 140
1 1 0 Homogeneous Effects 95 140
1 1 0 Heterogeneous Effects 100 140
Only Compensated
3 3 0 Homogeneous Effects 10 10
3 3 0 Heterogeneous Effects 10 15
2.5 3 0.5 Homogeneous Effects 10 15
2.5 3 0.5 Heterogeneous Effects 15 15
2.5 2.5 0 Homogeneous Effects 10 15
2.5 2.5 0 Heterogeneous Effects 15 15
2 3 1 Homogeneous Effects 15 20
2 3 1 Heterogeneous Effects 20 25
2 2.5 0.5 Homogeneous Effects 15 20
2 2.5 0.5 Heterogeneous Effects 20 25
2 2 0 Homogeneous Effects 15 20
2 2 0 Heterogeneous Effects 15 20
1.5 3 1.5 Homogeneous Effects 25 35
1.5 3 1.5 Heterogeneous Effects 30 45
1.5 2.5 1 Homogeneous Effects 25 30
1.5 2.5 1 Heterogeneous Effects 30 40
1.5 2 0.5 Homogeneous Effects 25 35
1.5 2 0.5 Heterogeneous Effects 25 35
1.5 1.5 0 Homogeneous Effects 25 35
1.5 1.5 0 Heterogeneous Effects 25 35
1 3 2 Homogeneous Effects 50 65
1 3 2 Heterogeneous Effects 70 95
1 2.5 1.5 Homogeneous Effects 50 70
1 2.5 1.5 Heterogeneous Effects 60 85
1 2 1 Homogeneous Effects 50 70
1 2 1 Heterogeneous Effects 60 80
1 1.5 0.5 Homogeneous Effects 50 70
1 1.5 0.5 Heterogeneous Effects 55 75
1 1 0 Homogeneous Effects 50 70
1 1 0 Heterogeneous Effects 50 70

Contour Plot

First let’s look at different deltadif values.

difindifsims_res_hom <- difindifsims_res %>% 
  filter(cv_delta==0) %>% 
  mutate(delta1 = as.numeric(as.character(delta1)),
         delta2 = as.numeric(as.character(delta2)),
         deltadif = as.numeric(as.character(deltadif))) %>% 
  mutate(power = ifelse(power == 1, 0.99, power)) 

difindifsims_res_het <- difindifsims_res %>% 
  filter(cv_delta!=0) %>% 
  mutate(delta1 = as.numeric(as.character(delta1)),
         delta2 = as.numeric(as.character(delta2)),
         deltadif = as.numeric(as.character(deltadif))) %>% 
  mutate(power = ifelse(power == 1, 0.99, power)) 

difindif_cont1_hom <- difindifsims_res_hom %>% 
  filter(deltadif!=3) %>% 
  ggplot(aes(x=n, y=delta1, z=power)) +
  geom_contour_filled(alpha=0.8, breaks=seq(0,1.1, by=0.1)) +
  facet_grid(deltadif~decomp, scales = "free") +
  theme_ipsum_rc() +
  labs(x = "Sample Size",
       y = "Hypothetical Intervention Effect of Main Treatment (mmHg)",
       subtitle = "Comparison of Differences of Effects between Interventions") +
  scale_y_continuous(breaks = seq(0.5, 5, by = 0.5)) +
  scale_fill_viridis("Power", discrete = T) +
  geom_contour(breaks=0.8, colour="black", linetype="dashed")

difindif_cont1_hom

difindif_cont1_het <- difindifsims_res_het %>% 
  filter(deltadif!=3) %>% 
  ggplot(aes(x=n, y=delta1, z=power)) +
  geom_contour_filled(alpha=0.8, breaks=seq(0,1.1, by=0.1)) +
  facet_grid(deltadif~decomp, scales = "free") +
  theme_ipsum_rc() +
  labs(x = "Sample Size",
       y = "Hypothetical Intervention Effect of Main Treatment (mmHg)",
       subtitle = "Comparison of Differences of Effects between Interventions") +
  scale_y_continuous(breaks = seq(0.5, 5, by = 0.5)) +
  scale_fill_viridis("Power", discrete = T) +
  geom_contour(breaks=0.8, colour="black", linetype="dashed")    

difindif_cont1_het

This is helpful to visualise, though probably not for the paper. With homogeneous effects, the determinant of the power is the difference in intervention effects; it’s a straight line. With heterogeneous effects, the line is not straight

difindif_cont2_hom <- ggplot(difindifsims_res_hom, aes(x=n, y=delta1, z=power)) +
  geom_contour_filled(alpha=0.8, breaks=seq(0,1.1, by=0.1)) +
  facet_grid(delta2~decomp, scales = "free") +
  theme_ipsum_rc() +
  labs(x = "Sample Size",
       y = "Hypothetical Intervention Effect of Main Treatment (mmHg)",
       subtitle = "Comparisons Grouped by Intervention Effects of the Reference Treatment (mmHg)") +
  scale_y_continuous(breaks = seq(0.5, 5, by = 0.5)) +
  scale_fill_viridis("Power", discrete = T) +
  geom_contour(breaks=0.8, colour="black", linetype="dashed")

difindif_cont2_hom

difindif_cont2_het <- ggplot(difindifsims_res_het, aes(x=n, y=delta1, z=power)) +
  geom_contour_filled(alpha=0.8, breaks=seq(0,1.1, by=0.1)) +
  facet_grid(delta2~decomp, scales = "free") +
  theme_ipsum_rc() +
  labs(x = "Sample Size",
       y = "Hypothetical Intervention Effect of Main Treatment (mmHg)",
       subtitle = "Comparisons Grouped by Intervention Effects of the Reference Treatment (mmHg)") +
  scale_y_continuous(breaks = seq(0.5, 5, by = 0.5)) +
  scale_fill_viridis("Power", discrete = T) +
  geom_contour(breaks=0.8, colour="black", linetype="dashed")    

difindif_cont2_het

And just looking against placebo (i.e. delta2=0)

difindif_cont3_hom <- difindifsims_res_hom %>% 
  filter(deltadif!=3) %>% 
  filter(delta2==0) %>% 
  ggplot(aes(x=n, y=delta1, z=power)) +
  geom_contour_filled(alpha=0.8, breaks=seq(0,1.1, by=0.1)) +
  facet_wrap(.~decomp, scales = "free") +
  theme_ipsum_rc() +
  labs(x = "Sample Size",
       y = "Hypothetical Intervention Effect of Main Treatment (mmHg)") +
  scale_y_continuous(breaks = seq(0.5, 5, by = 0.5)) +
  scale_fill_viridis("Power", discrete = T) +
  geom_contour(breaks=0.8, colour="black", linetype="dashed") +
  xlim(c(5,120))

difindif_cont3_hom

difindif_cont3_het <- difindifsims_res_het %>% 
  filter(deltadif!=3) %>% 
  filter(delta2==0) %>% 
  ggplot(aes(x=n, y=delta1, z=power)) +
  geom_contour_filled(alpha=0.8, breaks=seq(0,1.1, by=0.1)) +
  facet_wrap(.~decomp, scales = "free") +
  theme_ipsum_rc() +
  labs(x = "Sample Size",
       y = "Hypothetical Intervention Effect of Main Treatment (mmHg)") +
  scale_y_continuous(breaks = seq(0.5, 5, by = 0.5)) +
  scale_fill_viridis("Power", discrete = T) +
  geom_contour(breaks=0.8, colour="black", linetype="dashed") +
  xlim(c(5,120))

difindif_cont3_het

ggsave(difindif_cont3_hom, height=5, width=10, filename = "figures/Difindif_hom_contour.png")
ggsave(difindif_cont3_het, height=5, width=10, filename = "figures/Difindif_het_contour.png")

ggsave(difindif_cont3_hom, height=5, width=10, filename = "figures/Difindif_hom_contour.jpg", 
       dpi = 600)
ggsave(difindif_cont3_het, height=5, width=10, filename = "figures/Difindif_het_contour.jpg", 
       dpi = 600)

ICC Figures

colours <- c("#61b096", "#bd7969")

make_distributions <- function(icc) {
  
  sd_true <- 1
  var_true <- sd_true^2
  
  var_tot <- var_true / icc
  sd_tot <- sqrt(var_tot)
  
  sd_error <- sqrt(var_tot - var_true)
  
  distrib <- ggplot(data.frame(x=c(-5,5)), aes(x=x)) +
      stat_function(fun = dnorm, 
        colour = "black", size = 1.5, args = list(mean = 0, sd=sd_tot), 
        geom = "line") +
    stat_function(fun = dnorm, 
        colour = colours[1], size = 1, args = list(mean = 0, sd=sd_true), 
        geom = "line") +
    stat_function(fun = dnorm, 
        colour = colours[2], size = 1, args = list(mean = 0, sd=sd_error), 
        geom = "line") +
    geom_vline(xintercept = 0, linetype="dashed") +
    theme(axis.title.x = element_blank(),
          axis.title.y = element_blank(),
          axis.text.y=element_blank(),
          axis.ticks.y=element_blank(),
          axis.text.x=element_blank(),
          panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank()) +
    annotate("text", x=2.5, y=0.5, 
           label="Error\nVariance",
           colour = colours[2], hjust=0.5) +
    annotate("text", x=-2.5, y=0.5, 
           label="True\nVariance",
           colour = colours[1], hjust=0.5) +
    coord_cartesian(ylim=c(0,0.7))
  
  return(distrib)
}

make_piecharts <- function(icc) {
  
  sd_true <- 1
  var_true <- sd_true^2
  
  var_tot <- var_true / icc
  sd_tot <- sqrt(var_tot)
  
  var_error <- var_tot - var_true
  
  var_pie <- data.frame(Variance = c("True", "Error"),
                Values = c(var_true, var_error))
  var_pie$Variance <- forcats::fct_inorder(var_pie$Variance)
  
  
  pie <- ggplot(var_pie, aes(x="", y=Values, fill=Variance))+
    geom_bar(width = 1, stat = "identity") +
    theme(axis.text.y=element_blank(),
          axis.ticks.y=element_blank(),
          axis.text.x=element_blank(),
          axis.ticks.x=element_blank(),
          panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank()) +
    coord_polar("y", start=0) +
    scale_fill_manual(values = colours) +
    labs(x="", y="") +
    guides(fill=FALSE) +
    #labs(title=paste0("ICC=", icc)) +
    NULL
    
  
  return(pie)
}

make_distrib_and_pie <- function(icc) {
  
  dist <- make_distributions(icc)
  pie <- make_piecharts(icc)
  
  outplot <- cowplot::plot_grid(pie, dist, rel_widths = c(1,2)) +
    draw_figure_label(paste0("ICC=", icc))
  
  return(outplot)
  
}

comparison_charts <- tibble(
  icc = c(0.1, 0.3, 0.7, 0.9)
) %>% 
  mutate(chart = purrr::map(icc, make_distrib_and_pie))

plot_grid(plotlist = comparison_charts$chart, ncol=1)